The Hexablock: a domain associated with the $\mu$-synthesis in $M_2(\mathbb C)$

TL;DR

This paper introduces and analyzes the new hexablock domain in complex four-space, exploring its properties, boundary, automorphisms, and its relation to other domains, with applications to $mbda$-synthesis problems.

Contribution

The paper characterizes the hexablock domain, determines its boundary and automorphisms, and shows its relation to other key domains, providing new insights and alternative proofs in complex geometry.

Findings

Characterization of the hexablock domain

Determination of the distinguished boundary of bH

Establishment of the hexablock as an analytic retract of known domains

Abstract

We study a new domain in $\mathbb C^4$, namely the hexablock $\mathbb{H}$ that arises in connection with a special case of the $\mu$-synthesis problem in $M_2(\mathbb C)$. Previous attempts to study a few instances of the $\mu$-synthesis problem led to domains such as symmetrized bidisc $\mathbb G_2$, tetrablock $\mathbb E$ and pentablock $\mathbb P$. Throughout, the relations of $\mathbb{H}$ with these three domains are explored. Several characterizations of the hexablock are established. We determine the distinguished boundary $b \mathbb{H}$ of $\mathbb{H}$ and obtain a subgroup of the automorphism group of $\mathbb{H}$. The \textit{rational $\overline{\mathbb{H}}$-inner functions}, i.e., the rational functions from the unit disc $\mathbb{D}$ to $\mathbb{H}$ that map the boundary of $\mathbb{D}$ to $b\mathbb{H}$ are characterized. We obtain a Schwarz lemma for $\mathbb{H}$. Finally, it is shown that $\mathbb{G}_2$, $\mathbb{E}$, and $\mathbb{P}$ are analytic retracts of $\mathbb{H}$. By an application of the theory of the hexablock, we find alternative proofs to a few existing results on geometry and function theory of the pentablock.